An equivalent definition of the Betti tables of \(V\)  is based on its Koszul complex. Given an \(n\) -graded \(S\) -module \(V=\bigoplus _{u\in \mathbb {Z}^n}V^u\) , the Koszul complex of \(V\)  at grade \(u\in \mathbb {Z}^n\), denoted \(\mathbb {K}_*(x_1,\ldots ,x_n; V)(u)\) , is the part of grade \(u\)  of the (\(n\) -graded) chain complex \(V \otimes _S \mathbb {K}_*\) , where \(\mathbb {K}_*= \mathbb {K}_*(x_1,\ldots ,x_n)\)  is the classical Koszul complex of \(S\) , defined for example in [1]} or [2]}.
Below, we provide an explicit definition of \(\mathbb {K}_*(x_1,\ldots ,x_n; V)(u)\) . Since \(\mathbb {K}_*\)  is a minimal free resolution of \(\mathbb {F}\cong S/\langle x_1, \ldots ,x_n\rangle \)  [1]}, it is easy to show (see, e.g., [4]}) that the \(i\) th homology module of the chain complex \(V \otimes _S \mathbb {K}_*\)  has dimensions (over \(\mathbb {F}\) ) in the various grades \(u\in \mathbb {Z}^n\)  coinciding with the Betti table \(\xi _i\)  of \(V\) . In other words, for each \(u\in \mathbb {Z}^n\) ,
\(\xi _i (u) = \dim _{\mathbb {F}} H_i (\mathbb {K}_*(x_1,\ldots ,x_n; V)(u)) .\) 
